NORWEGIAN UNIVERSITYOF SCIENCE ANDTECHNOLOGY
Department of Physics
Contacts duringtheexam:
ProfessorJanMyrheim, tel.73 59 3653
ProfessorPerHemmer, tel.73 5936 48
74310 QUANTUM MECHANICS 1
Tuesday June 1, 1999
09.00 {15.00
Allowed aids: Acceptable calculator
Rottmann: Matematisk formelsamling
Three pages with expressionsand formulae are enclosed.
Examinationresults willbe availableon June 22,1999.
Problem 1
a) Aparticle with mass m is inthe Coulombpotential
V(r)= e
2
4"
0 r
:
What is the time-dependent Schrodinger equation for the particle?
What is meant by aquantum-mechanicalstationary state?
At time t=0the wave functionof the particleis
(~r;0)=(3a 3
0 )
1
2
e r=a
0
+(48a 5
0 )
1
2
re r=(2a
0 )
cos#; (1)
where a
0
is the Bohr radius.
Is this wave functiona stationarystate?
b) An energy measurement is performed when the particle is in the state (1). What are
the possibleresults, and what are their probabilities?
c) The particleis then in the ground state in the Coulomb potential,with energy E 0
1 . A
weak constant electriceld E in the z direction, corresponding tothe potential
H 0
= eEz;
is applied. Here e is the charge of the particle. Assume that for a weak eld the resulting
ground-state energy may be expanded as apowers of the eld strength:
E
1
(E)=E 0
1
+AE+BE 2
+::: (2)
This is the Stark eect.
Use stationary perturbationtheory tocompute A, and to predictthe sign of B.
d) What are the possible results one may obtainif one (for ageneral state) measures the
square
~
L 2
and the componentL
z
of the angular momentum?
LetthentheparticleintheCoulombpotentialatt=0havethefollowingwavefunction
(~r)=
1
q
a 3
0 (1+
2
)
1+ z
a
0
e r=a
0
; (3)
with a mean energy
hjH
0 ji=
h 2
2ma 2
0 (1+
2
) :
(This result is given for later use, proof is not required.) Here is a real parameter, and
H
0
isthe Hamiltonianof the particle.
What are the possible measurement results for
~
L 2
and L
z
for a particle inthe special
state (3)?
e)Showthat theground-stateenergy E
1
foraparticlewithHamiltonianoperatorHnever
exceeds the Rayleigh-Ritz estimate
E
RR
=hfjHjfi= Z
f
(~r)H f(~r)d 3
~r;
for any function f(~r)that is normalized:
hfjfi= Z
jf(~r)j 2
d 3
~r=1:
f)Usethefunction(~r),equation(3),asatrialfunctiontoobtainaRayleigh-Ritzestimate
for the Stark eect in the ground state in Coulomb potential, i.e. for the Hamiltonian
operator
H =H
0
eEz:
It is the coeÆcient B inpower series (2) that is tobe determined.
Since (~r) with parameter = 0 is the ground-state wave function with E = 0, is
necessarilysmallwhentheelectriceldisweak. Youmaythereforesimplifythecalculation
by expanding tosecond order in .
g) The electron has spin 1=2. The total angular momentum of an electron is
~
J =
~
L+
~
S,
where
~
L=~r~pis the orbitalangularmomentum and
~
S isthe spin. Denote the quantum
numbers for
~
L 2
,L
z ,
~
S 2
, S
z ,
~
J 2
and J
z
by l,m
l
=m, s=1=2, m
s
, j and m
j
, respectively.
Assume now that l =1. Whichvalues are then possible for j and m
j
?
Showthat the state
j i= q
2
3
l=1;s= 1
2
;m
l
=1;m
s
= 1
2 E
q
1
3
l=1;s= 1
2
;m
l
=0;m
s
= 1
2 E
is aneigenstate for
~
J 2
and J
z
, and nd the eigenvalues. Use, e.g., the relations
~
J 2
=
~
L 2
+
~
S 2
+2
~
L
~
S =
~
L 2
+
~
S 2
+L
+
S +L S
+ +2L
z S
z
;
where L
=L
x iL
y , S
=S
x iS
y , and
L
jl;s;m
l
;m
s i =h
q
l(l+1) m
l (m
l
1) jl;s;m
l
1;m
s i;
S
jl;s;m
l
;m
s i =h
q
s(s+1) m
s (m
s
1) jl;s;m
l
;m
s 1i:
In this state j i, what isthe probability that m
s
=1=2?
Use the attached tables of Clebsch{Gordan-coeÆcients, and express the state with
l =1, s=1=2, m
l
=0 and m
s
=1=2 asa linear combinationof states with dierent j.
In this last state, what isthe probability that j =1=2?
h)Relativisticcorrections (spin-orbitcoupling) and correctionsdue toquantization ofthe
electromagnetic eldmakethe energy levelsof the hydrogen atom dependentonthe main
quantum number n, the orbital angular momentum l and in addition on the quantum
number j of
~
J 2
= (
~
L+
~
S) 2
. These corrections lift the degeneracy between energy levels
withthe samemainquantum numbernbut withdierentvaluesofl. Wewillneglecthere
alleects havingto dowith the electron spin, except for the energy splittingbetween the
energy levels2s og2p. LetE denotethe energy dierence, sothat E=E
2s E
2p .
Whatspontaneoustransitionsbetweenthestates2s,2pand1scantakeplacebyelectric
i) Assume here that the 2s level lies above the 2p level, and that the energy dierence
(known as the Lamb shift)is
E
L
=h!
L
=4;3810 6
eV:
Atransitionfromthe2sleveltothe2plevelmaybeinducedbyanoscillatinghomogeneous
electriceld, given by the potential
U(z;t)= eEzsin(!t);
where E and ! are constants.
Use rst order time dependent perturbation theory to compute the probability for an
induced transitionfrom 2s att =0 to 2patt =T. Neglect the electron spin, and assume
that the nal state has m
l
=m =0.
Howlarge must E bein order that the transition probabilty is1% duringone second,
when !=!
L
?
Is therereason to believe that the rst order approximation is good inthis case?
Energy eigenvalues and eigenfunctions in the Coulomb potential
E 0
n
= m
2h 2
e 2
4"
0
!
2
1
n 2
= h 2
2ma 2
0 1
n 2
:
n l
nl m
1 0
100
= (a 3
) 1
2
e r=a
1s
0
200
= (32a 3
) 1
2
(2 r=a)e r=(2a)
2s
2
210
= (32a 5
) 1
2
r e r=(2a)
cos#
1
211
= (64a 5
) 1
2
r e r=(2a)
sin#e i'
2p
21 1
= (64a 5
) 1
2
r e r=(2a)
sin#e i'
Physical constants
The speed of light c=2:997910 8
m/s
Planck's constant h=2h=6:626210 34
Js
The electron charge e jej=1:602210 19
C
The electron mass m
e
=9:109610 31
kg
The ne structure constant =e 2
=(4"
0
hc)=1=137:05
The Bohr radius a
0
=4"
0 h 2
=(m
e e
2
)
Time independent perturbation theory
For H =H 0
+H
1
we have
E
n
=E 0
n
+hnjH
1 jni+
X
m(6=n)
jhmjH
1 jnij
2
E 0
n E
0
m
+O(
3
):
Integrals
Z
1
0 e
x 2
dx= 1
2 p
Z
1
0 t
n
e t
dt=n!
The Laplace operator in spherical coordinates
r 2
=
@ 2
@r 2
+ 2
r
@
@r +
1
r 2
"
@ 2
@#
2
+cot#
@
@#
+ 1
sin 2
#
@ 2
@' 2
#
Orbital angular momentum
In sphericalcoordinates:
~
L 2
= h 2
"
@ 2
@# 2
+cot#
@
@#
+ 1
sin 2
#
@ 2
@' 2
#
Eigenvalues:
~
L 2
Y
l m
(#;') = l(l+1)h 2
Y
l m (#;')
L
z Y
l m
(#;') = mh Y
l m (#;')
Time dependent perturbation theory
In rstorder timedependentperturbation theorythe probabilityamplitudefora tran-
sition from a state
i
at time t =0 to a state
f
at t = T, induced by a time dependent
perturbing potentialV(~r;t), is
a
i!f
= 1
ih Z
T
0 dtV
fi (t)e
i!
fi t
:
i and
f
are eigenstates of the unperturbed Hamiltonianoperator with energies E
i and
E
f
, respectively. Furthermore, !
fi
=(E
f E
i
)=h, and
V
fi (t)=
Z
d 3
~ r
f
(~r)V(~r;t)
i (~r):
In the electric dipoleapproximation the probabilitypertime forspontaneous emission
of electromagneticradiationwithinaninnitesimalsolid angled andwith apolarisation
vector~, with j~j=1, is
d= !
3
2c 2
~
~
d
2
d:
Here ! =(E
i E
f
)=h, is the ne structure constant, and
~
d= Z
d 3
~r
f
~r
i :
